The Mathematics of GeoEnergy

The Oil Shock Model and Compartmental Models

Chapter 5 of the book describes a model of the production of oil based on discoveries followed by a sequence of lags relating to decisions made and physical constraints governing the flow of that oil. As it turns out, this so-named Oil Shock Model is mathematically similar to the compartmental models used to model contagion growth in epidemiology, pharmaceutical/drug deliver systems, and other applications as demonstrated in Appendix E of the book.

One aspect of the 2020 pandemic is that everyone with any math acumen is becoming aware of contagion models such as the SIR compartmental model, where S I R stands for Susceptible, Infectious, and Recovered individuals. The Infectious part of the time progression within a population resembles a bell curve that peaks at a particular point indicating maximum contagiousness. The hope is that this either peaks quickly or that it doesn’t peak at too high a level.

In the modeling of oil depletion the equivalent S I R compartmental model corresponds to the stages of Sequestered (in the ground), Identified (i.e. discovered), and Recovered (i.e. extracted). So the compartmental models of oil production and contagion growth (as it plays out during a pandemic) show intuitive parallels :

Discovery of an oil reservoir is analogous to the start of infection — the leading indicator.

Extraction from that reservoir to depletion is analogous to death — the lagging indicator.

Keeping oil in the ground is the equivalent of recovering from the infection.

This has been progressing over the course of decades, with the global peak of the discovered oil occurring by the end of the 1960’s and on a downhill trajectory since then — a slow but relentless extraction drawdown with the majority of the citizenry barely being aware of this trend. The derivative of the recovery phase — the extraction — is shown below and is either at peak or near it.

This slow progression of global oil production is nowhere near as sudden as what we’re going through now as the full S I R coronavirus cycle completes in a matter of months. And this virus cycle may recur again, whereas the S I R version for oil will not — as oil does not regenerate.

Since the connection between the compartmental model of the oil shock model and the pandemic compartmental model is so striking, there are other ways to analogize between the two. Quarantining people from accessing the oil during the oil embargo of the 1970’s (see the figure above for the notch right before 1980) produces the same effect as quarantining during an epidemic (see the notch in the figure below).

When the quarantining policies are relaxed, the contagion will resume — the pattern is spike, suppress, resume. From our book, in the figure below, one can see how the “quarantining=embargo” suppression flattened the production curve below what it was likely trending toward, thus temporarily delaying the time to full depletion.

The analogy is that people are the contagion and will continue to extract oil until the equivalent herd immunity is reached — that is until when the finite oil resources are exhausted. Only an embargo shock prevented the consumption from proceeding too quickly, without which the supply would deplete much sooner.

Another analogy is in how dispersion works. When discovering oil, not every geographic location is exploited at the same time, leading to a spread in how quickly the reserves are exhausted. This is a stochastic element to the compartmental model, the figure below taken from Chapter 7 with the stochastic version of the logistic S-curve discussed recently at the Azimuth Project forum.

As stated earlier, the problem with contagions is that the S I R process can potentially repeat, as the populations are renewable. This recurrence is not possible with crude oil, where the supply is finite & non-renewable. While this is strictly true, the finiteness can be masked if there are hidden pockets of reserves, such as with the shale oil reserves of the Permian and Bakken formations. This can be seen in the USA production data shown below, where a spike recovery in production occurred starting in 2010 (the costly fracking of oil partly financed by government stimulation after the 2008-2009 market crash).

This is weakly analogous to a reemergence of a contagion. However, this only provides a temporary reprieve to global oil depletion as the shale oil is already proving to be near peak.

What does the global economy look forward to? Even when we eventually contain the pandemic, the inexorable decline of oil will continue. The pandemic itself will provide a transient suppressive shock to global oil production — via demand destruction of production this only temporarily delays the inevitability of continued oil depletion.

Even though the idea of compartmental modeling is well-known in epidemiology, one can find little evidence as to its application to fossil fuel depletion. Google Scholar citations for “compartmental model” & “oil depletion”

The way that Herroro et al apply a compartmental model is to model the compartments as oil, atmospheric CO2 from combustion of the oil, and sequestering of that CO2 in the ocean. We also model this compartmental flow in Chap 9.

As of April 1, there are many burgeoning collective efforts on optimizing pandemic contagion S I R models, as listed here (and I contributed a version of Hubbert Linearization for pandemic modeling on Medium.com). Perhaps eventually, the same level of attention will be paid to fossil fuel depletion compartmental models such as the oil shock model, as the impact on society is comparable.